The public transit system used to be privately owned and operated. Then the cities took over and kicked out the private sector. Now we have terrible public transit. Yes, even the subways in NYC were built and operated by private companies (there were two, and they competed).
So, I don't think you can blame the shift from private to public for this. It's not that black and white. I think the car lobby has been a much bigger factor in driving both the construction of highways, suburbs, and destruction of local tram networks across most cities in the US. This did not happen in Europe.
I was in San Francisco a few months ago. The public transport there is pretty bad compared to what I'm used to in Berlin (clean, fast, efficient, etc.). The local muni trains are tiny and infrequent and you basically have to step over hords of homeless people to get into them. I was kind of shocked to see a 1 car muni show up after waiting for that for close to 15 minutes at Powel station. It's like they're not even trying.